text:
Exponentiation
Cantor normal form
Factorization into primes
Large countable ordinals
Natural operations
Nimber arithmetic
Ordinal arithmetic
Operations on ordinals that extend classical arithmetic
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations.
The union of two disjoint well-ordered sets S and T can be well-ordered. The order-type of that union is the ordinal that results from adding the order-types of S and T. If two well-ordered sets are not already disjoint, then they can be replaced by order-isomorphic disjoint sets, e.g. replace S by {0} × S and T by {1} × T. This way, the well-ordered set S is written "to the left" of the well-ordered set T, meaning one defines an order on S ∪ {\displaystyle \cup } T in which every element of S is smaller than every element of T. The sets S and T themselves keep the ordering they already have. This addition of the order-types is associative and generalizes the addition of natural numbers.
The definition of addition can also be given inductively (the following induction is on β):
α + 0 = α,
α + (β + 1) = (α + β) + 1 (here, "+ 1" denotes the successor of an ordinal),
and if β is a limit ordinal then α + β is the limit of α + δ for all δ < β.
The first transfinite ordinal is ω, the set of all natural numbers. For example, the ordinal ω + ω is obtained by two copies of the natural numbers ordered in the usual fashion and the second copy completely to the right of the first. Writing 0' < 1' < 2' < ... for the second copy, ω + ω looks like
0 < 1 < 2 < 3 < ... < 0' < 1' < 2' < ...
This is different from ω because in ω only 0 does not have a direct predecessor while in ω + ω the two elements 0 and 0' do not have direct predecessors. As another example, here are 3 + ω and ω + 3:
0 < 1 < 2 < 0' < 1' < 2' < ...
0 < 1 < 2 < ... < 0' < 1' < 2'
After relabeling, the former just looks like ω itself, i.e. 3 + ω = ω, while the latter does not: ω + 3 is not equal to ω since ω + 3 has a largest element (namely, 2') and ω does not (even if ω and ω + 3 are equipotent, they are not isomorphic). Hence, this addition is not commutative. In fact it is quite rare for α+β to be equal to β+α: this happens if and only if α=γm, β=γn for some ordinal γ and natural numbers m and n. From this it follows that "α commutes with β" is an equivalence relation on the class of nonzero ordinals, and all the equivalence classes are countably infinite. However, addition is still associative; one can see for example that (ω + 4) + ω = ω + (4 + ω) = ω + ω.
Addition is strictly increasing and continuous in the right argument:
α < β ⇒ γ + α < γ + β {\displaystyle \alpha <\beta \Rightarrow \gamma +\alpha <\gamma +\beta }
but the analogous relation does not hold for the left argument; instead we only have:
α < β ⇒ α + γ ≤ β + γ {\displaystyle \alpha <\beta \Rightarrow \alpha +\gamma \leq \beta +\gamma }
Ordinal addition is left-cancellative: if α + β = α + γ, then β = γ. Furthermore, one can define left subtraction for ordinals β ≤ α: there is a unique γ such that α = β + γ. On the other hand, right cancellation does not work:
3 + ω = 0 + ω = ω {\displaystyle 3+\omega =0+\omega =\omega } but 3 ≠ 0 {\displaystyle 3\neq 0}
Nor does right subtraction, even when β ≤ α: for example, there does not exist any γ such that γ + 42 = ω.
If the ordinals less than α are closed under addition and contain 0 then α is occasionally called a γ-number (see additively indecomposable ordinal). These are exactly the ordinals of the form ωβ.
The Cartesian product, S×T, of two well-ordered sets S and T can be well-ordered by a variant of lexicographical order that puts the least significant position first. Effectively, each element of T is replaced by a disjoint copy of S. The order-type of the Cartesian product is the ordinal that results from multiplying the order-types of S and T. Again, this operation is associative and generalizes the multiplication of natural numbers.
Here is ω·2:
00 < 10 < 20 < 30 < ... < 01 < 11 < 21 < 31 < ...,
which has the same order type as ω + ω. In contrast, 2·ω looks like this:
00 < 10 < 01 < 11 < 02 < 12 < 03 < 13 < ...
and after relabeling, this looks just like ω. Thus, ω·2 = ω+ω ≠ ω = 2·ω, showing that multiplication of ordinals is not commutative. More generally, a natural number greater than 1 never commutes with any infinite ordinal, and two infinite ordinals α, β commute if and only if αm = βn for some positive natural numbers m and n. The relation "α commutes with β" is an equivalence relation on the ordinals greater than 1, and all equivalence classes are countably infinite.
Distributivity partially holds for ordinal arithmetic: α(β+γ) = αβ+αγ. However, the other distributive law (β+γ)α = βα+γα is not generally true: (1+1)·ω = 2·ω = ω while 1·ω+1·ω = ω+ω, which is different. Therefore, the ordinal numbers form a left near-semiring, but do not form a ring.
The definition of multiplication can also be given inductively (the following induction is on β):
α·0 = 0,
α·(β+1) = (α·β)+α,
and if β is a limit ordinal then α·β is the limit of the α·δ for δ < β.
The main properties of the product are:
α·0 = 0·α = 0.
One (1) is a multiplicative identity α·1 = 1·α = α.
Multiplication is associative (α·β)·γ = α·(β·γ).
Multiplication is strictly increasing and continuous in the right argument: (α < β and γ > 0) ⇒ {\displaystyle \Rightarrow } γ·α < γ·β
Multiplication is not strictly increasing in the left argument, for example, 1 < 2 but 1·ω = 2·ω = ω. However, it is (non-strictly) increasing, i.e. α ≤ β ⇒ {\displaystyle \Rightarrow } α·γ ≤ β·γ.
There is a left cancellation law: If α > 0 and α·β = α·γ, then β = γ.
Right cancellation does not work, e.g. 1·ω = 2·ω = ω, but 1 and 2 are different.
α·β = 0 ⇒ {\displaystyle \Rightarrow } α = 0 or β = 0.
Distributive law on the left: α·(β+γ) = α·β+α·γ
No distributive law on the right: e.g. (ω+1)·2 = ω+1+ω+1 = ω+ω+1 = ω·2+1, which is not ω·2+2.
Left division with remainder: for all α and β, if β > 0, then there are unique γ and δ such that α = β·γ+δ and δ < β. (This does not however mean the ordinals are a Euclidean domain, since they are not even a ring, and the Euclidean "norm" is ordinal-valued.)
Right division does not work: there is no α such that α·ω ≤ ωω ≤ (α+1)·ω.
A δ-number (see Multiplicatively indecomposable) is an ordinal greater than 1 such that αβ=β whenever 0<α<β. These consist of the ordinal 2 and the ordinals of the form ωωβ.
The definition of ordinal exponentiation for finite exponents is straightforward. If the exponent is a finite number, the power is the result of iterated multiplication. For instance, ω2 = ω·ω using the operation of ordinal multiplication. Note that ω·ω can be defined using the set of functions from 2 = {0,1} to ω = {0,1,2,...}, ordered lexicographically with the least significant position first:
(0,0) < (1,0) < (2,0) < (3,0) < ... < (0,1) < (1,1) < (2,1) < (3,1) < ... < (0,2) < (1,2) < (2,2) < ...
Here for brevity, we have replaced the function {(0,k), (1,m)} by the ordered pair (k, m).
Similarly, for any finite exponent n, ω n {\displaystyle \omega ^{n}} can be defined using the set of functions from n (the domain) to the natural numbers (the codomain). These functions can be abbreviated as n-tuples of natural numbers.
But for infinite exponents, the definition may not be obvious. A limit ordinal, such as ωω, is the supremum of all smaller ordinals. It might seem natural to define ωω using the set of all infinite sequences of natural numbers. However, we find that any absolutely defined ordering on this set is not well-ordered.[1] To deal with this issue we can use the variant lexicographical ordering again. We restrict the set to sequences that are nonzero for only a finite number of arguments. This is naturally motivated as the limit of the finite powers of the base (similar to the concept of coproduct in algebra). This can also be thought of as the infinite union ⋃ n < ω ω n {\displaystyle \bigcup _{n<\omega }\omega ^{n}} .
Each of those sequences corresponds to an ordinal less than ω ω {\displaystyle \omega ^{\omega }} such as ω n 1 c 1 + ω n 2 c 2 + ⋯ + ω n k c k {\displaystyle \omega ^{n_{1}}c_{1}+\omega ^{n_{2}}c_{2}+\cdots +\omega ^{n_{k}}c_{k}} and ω ω {\displaystyle \omega ^{\omega }} is the supremum of all those smaller ordinals.
The lexicographical order on this set is a well ordering that resembles the ordering of natural numbers written in decimal notation, except with digit positions reversed, and with arbitrary natural numbers instead of just the digits 0–9:
(0,0,0,...) < (1,0,0,0,...) < (2,0,0,0,...) < ... <
(0,1,0,0,0,...) < (1,1,0,0,0,...) < (2,1,0,0,0,...) < ... <
(0,2,0,0,0,...) < (1,2,0,0,0,...) < (2,2,0,0,0,...)
< ... <
(0,0,1,0,0,0,...) < (1,0,1,0,0,0,...) < (2,0,1,0,0,0,...)
< ...
In general, any ordinal α can be raised to the power of another ordinal β in the same way to get αβ.
It is easiest to explain this using Von Neumann's definition of an ordinal as the set of all smaller ordinals. Then, to construct a set of order type αβ consider all functions from β to α such that only a finite number of elements of the domain β map to a non zero element of α (essentially, we consider the functions with finite support). The order is lexicographic with the least significant position first. We find
1ω = 1,
2ω = ω,
2ω+1 = ω·2 = ω+ω.
The definition of exponentiation can also be given inductively (the following induction is on β, the exponent):
α0 = 1,
αβ+1 = (αβ)·α, and
if β is a limit ordinal, then αβ is the limit of the αδ for all δ < β.
Properties of ordinal exponentiation:
α0 = 1.
If 0 < α, then 0α = 0.
1α = 1.
α1 = α.
αβ·αγ = αβ + γ.
(αβ)γ = αβ·γ.
There are α, β, and γ for which (α·β)γ ≠ αγ·βγ. For instance, (ω·2)2 = ω·2·ω·2 = ω2·2 ≠ ω2·4.
Ordinal exponentiation is strictly increasing and continuous in the right argument: If γ > 1 and α < β, then γα < γβ.
If α < β, then αγ ≤ βγ. Note, for instance, that 2 < 3 and yet 2ω = 3ω = ω.
If α > 1 and αβ = αγ, then β = γ. If α = 1 or α = 0 this is not the case.
For all α and β, if β > 1 and α > 0 then there exist unique γ, δ, and ρ such that α = βγ·δ + ρ such that 0 < δ < β and ρ < βγ.
While the same notation is used for ordinal exponentiation and cardinal exponentiation, ordinal exponentiation is quite different from cardinal exponentiation. For example, with ordinal exponentiation 2 ω = ω {\displaystyle 2^{\omega }=\omega } , but for ℵ 0 {\displaystyle \aleph _{0}} (aleph naught, the cardinality of ω {\displaystyle \omega } ), 2 ℵ 0 > ℵ 0 {\displaystyle 2^{\aleph _{0}}>\aleph _{0}} . Here, 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} is the cardinality of the set of all functions from the set of all natural numbers to a set with two elements. (This is the cardinality of the power set of the set of all natural numbers and is equal to c {\displaystyle {\mathfrak {c}}} , the cardinality of the continuum.) To avoid confusing ordinal exponentiation with cardinal exponentiation, one can use symbols for ordinals (e.g. ω) in the former and symbols for cardinals (e.g. ℵ 0 {\displaystyle \aleph _{0}} ) in the latter.
Jacobsthal showed that the only solutions of αβ = βα with α ≤ β are given by α = β, or α = 2 and β = 4, or α is any limit ordinal and β = εα where ε is an ε-number larger than α.[2]
Every ordinal number α can be uniquely written as ω β 1 c 1 + ω β 2 c 2 + ⋯ + ω β k c k {\displaystyle \omega ^{\beta _{1}}c_{1}+\omega ^{\beta _{2}}c_{2}+\cdots +\omega ^{\beta _{k}}c_{k}} , where k is a natural number, c 1 , c 2 , … , c k {\displaystyle c_{1},c_{2},\ldots ,c_{k}} are positive integers, and β 1 > β 2 > … > β k ≥ 0 {\displaystyle \beta _{1}>\beta _{2}>\ldots >\beta _{k}\geq 0} are ordinal numbers. The degenerate case α=0 occurs when k=0 and there are no βs nor cs. This decomposition of α is called the Cantor normal form of α, and can be considered the base-ω positional numeral system. The highest exponent β 1 {\displaystyle \beta _{1}} is called the degree of α {\displaystyle \alpha } , and satisfies β 1 ≤ α {\displaystyle \beta _{1}\leq \alpha } . The equality β 1 = α {\displaystyle \beta _{1}=\alpha } applies if and only if α = ω α {\displaystyle \alpha =\omega ^{\alpha }} . In that case Cantor normal form does not express the ordinal in terms of smaller ones; this can happen as explained below.
A minor variation of Cantor normal form, which is usually slightly easier to work with, is to set all the numbers ci equal to 1 and allow the exponents to be equal. In other words, every ordinal number α can be uniquely written as ω β 1 + ω β 2 + ⋯ + ω β k {\displaystyle \omega ^{\beta _{1}}+\omega ^{\beta _{2}}+\cdots +\omega ^{\beta _{k}}} , where k is a natural number, and β 1 ≥ β 2 ≥ … ≥ β k ≥ 0 {\displaystyle \beta _{1}\geq \beta _{2}\geq \ldots \geq \beta _{k}\geq 0} are ordinal numbers.
Another variation of the Cantor normal form is the "base δ expansion", where ω is replaced by any ordinal δ>1, and the numbers ci are positive ordinals less than δ.
The Cantor normal form allows us to uniquely express—and order—the ordinals α that are built from the natural numbers by a finite number of arithmetical operations of addition, multiplication and exponentiation base- ω {\displaystyle \omega } : in other words, assuming β 1 < α {\displaystyle \beta _{1}<\alpha } in the Cantor normal form, we can also express the exponents β i {\displaystyle \beta _{i}} in Cantor normal form, and making the same assumption for the β i {\displaystyle \beta _{i}} as for α and so on recursively, we get a system of notation for these ordinals (for example,
ω ( ω ( ω 7 ⋅ 6 + ω + 42 ) ⋅ 1729 + ω 9 + 88 ) ⋅ 3 + ω ( ω ω ) ⋅ 5 + 65537 {\displaystyle \omega ^{\left(\omega ^{\left(\omega ^{7}\cdot 6+\omega +42\right)}\cdot 1729+\omega ^{9}+88\right)}\cdot 3+\omega ^{\left(\omega ^{\omega }\right)}\cdot 5+65537}
denotes an ordinal).
The ordinal ε0 (epsilon nought) is the set of ordinal values α of the finite-length arithmetical expressions of Cantor normal form that are hereditarily non-trivial where non-trivial means β1<α when 0<α. It is the smallest ordinal that does not have a finite arithmetical expression in terms of ω, and the smallest ordinal such that ε 0 = ω ε 0 {\displaystyle \varepsilon _{0}=\omega ^{\varepsilon _{0}}} , i.e. in Cantor normal form the exponent is not smaller than the ordinal itself. It is the limit of the sequence
0 , 1 = ω 0 , ω = ω 1 , ω ω , ω ω ω , … . {\displaystyle 0,\,1=\omega ^{0},\,\omega =\omega ^{1},\,\omega ^{\omega },\,\omega ^{\omega ^{\omega }},\,\ldots \,.}
The ordinal ε0 is important for various reasons in arithmetic (essentially because it measures the proof-theoretic strength of the first-order Peano arithmetic: that is, Peano's axioms can show transfinite induction up to any ordinal less than ε0 but not up to ε0 itself).
The Cantor normal form also allows us to compute sums and products of ordinals: to compute the sum, for example, one need merely know (see the properties listed in § Addition and § Multiplication) that
ω β c + ω β ′ c ′ = ω β ′ c ′ , {\displaystyle \omega ^{\beta }c+\omega ^{\beta '}c'=\omega ^{\beta '}c'\,,}
if β ′ > β {\displaystyle \beta '>\beta } (if β ′ = β {\displaystyle \beta '=\beta } one can apply the distributive law on the left and rewrite this as ω β ( c + c ′ ) {\displaystyle \omega ^{\beta }(c+c')} , and if β ′ < β {\displaystyle \beta '<\beta } the expression is already in Cantor normal form); and to compute products, the essential facts are that when 0 < α = ω β 1 c 1 + ⋯ + ω β k c k {\displaystyle 0<\alpha =\omega ^{\beta _{1}}c_{1}+\cdots +\omega ^{\beta _{k}}c_{k}} is in Cantor normal form and 0 < β ′ {\displaystyle 0<\beta '} , then
α ω β ′ = ω β 1 + β ′ {\displaystyle \alpha \omega ^{\beta '}=\omega ^{\beta _{1}+\beta '}\,}
α n = ω β 1 c 1 n + ω β 2 c 2 + ⋯ + ω β k c k , {\displaystyle \alpha n=\omega ^{\beta _{1}}c_{1}n+\omega ^{\beta _{2}}c_{2}+\cdots +\omega ^{\beta _{k}}c_{k}\,,}
if n is a non-zero natural number.
To compare two ordinals written in Cantor normal form, first compare β 1 {\displaystyle \beta _{1}} , then c 1 {\displaystyle c_{1}} , then β 2 {\displaystyle \beta _{2}} , then c 2 {\displaystyle c_{2}} , etc.. At the first difference, the ordinal that has the larger component is the larger ordinal. If they are the same until one terminates before the other, then the one that terminates first is smaller.
Ernst Jacobsthal showed that the ordinals satisfy a form of the unique factorization theorem: every nonzero ordinal can be written as a product of a finite number of prime ordinals. This factorization into prime ordinals is in general not unique, but there is a "minimal" factorization into primes that is unique up to changing the order of finite prime factors (Sierpiński 1958).
A prime ordinal is an ordinal greater than 1 that cannot be written as a product of two smaller ordinals. Some of the first primes are 2, 3, 5, ... , ω, ω+1, ω2+1, ω3+1, ..., ωω, ωω+1, ωω+1+1, ... There are three sorts of prime ordinals:
The finite primes 2, 3, 5, ...
The ordinals of the form ωωα for any ordinal α. These are the prime ordinals that are limits, and are the delta numbers.
The ordinals of the form ωα+1 for any ordinal α>0. These are the infinite successor primes, and are the successors of gamma numbers, the additively indecomposable ordinals.
Factorization into primes is not unique: for example, 2×3=3×2, 2×ω=ω, (ω+1)×ω=ω×ω and ω×ωω = ωω. However, there is a unique factorization into primes satisfying the following additional conditions:
Every limit prime occurs before every successor prime
If two consecutive primes of the prime factorization are both limits or both finite, then the second one is at most the first one.
This prime factorization can easily be read off using the Cantor normal form as follows:
First write the ordinal as a product αβ where α is the smallest power of ω in the Cantor normal form and β is a successor.
If α=ωγ then writing γ in Cantor normal form gives an expansion of α as a product of limit primes.
Now look at the Cantor normal form of β. If β = ωλm + ωμn + smaller terms, then β = (ωμn + smaller terms)(ωλ−μ + 1)m is a product of a smaller ordinal and a prime and an integer m. Repeating this and factorizing the integers into primes gives the prime factorization of β.
So the factorization of the Cantor normal form ordinal
ω α 1 n 1 + ⋯ + ω α k n k {\displaystyle \omega ^{\alpha _{1}}n_{1}+\cdots +\omega ^{\alpha _{k}}n_{k}} (with α 1 > ⋯ > α k {\displaystyle \alpha _{1}>\cdots >\alpha _{k}} )
into a minimal product of infinite primes and integers is
ω ω β 1 ⋯ ω ω β m n k ( ω α k − 1 − α k + 1 ) n k − 1 ⋯ n 2 ( ω α 1 − α 2 + 1 ) n 1 {\displaystyle \omega ^{\omega ^{\beta _{1}}}\cdots \omega ^{\omega ^{\beta _{m}}}n_{k}(\omega ^{\alpha _{k-1}-\alpha _{k}}+1)n_{k-1}\cdots n_{2}(\omega ^{\alpha _{1}-\alpha _{2}}+1)n_{1}}
where each ni should be replaced by its factorization into a non-increasing sequence of finite primes and
α k = ω β 1 + ⋯ + ω β m {\displaystyle \alpha _{k}=\omega ^{\beta _{1}}+\cdots +\omega ^{\beta _{m}}} with β 1 ≥ ⋯ ≥ β m {\displaystyle \beta _{1}\geq \cdots \geq \beta _{m}} .
As discussed above, the Cantor normal form of ordinals below ε 0 {\displaystyle \varepsilon _{0}} can be expressed in an alphabet containing only the function symbols for addition, multiplication and exponentiation, as well as constant symbols for each natural number and for ω {\displaystyle \omega } . We can do away with the infinitely many numerals by using just the constant symbol 0 and the operation of successor, S {\displaystyle S} (for example, the integer 4 may be expressed as S ( S ( S ( S ( 0 ) ) ) ) {\displaystyle S(S(S(S(0))))} ). This describes an ordinal notation: a system for naming ordinals over a finite alphabet. This particular system of ordinal notation is called the collection of arithmetical ordinal expressions, and can express all ordinals below ε 0 {\displaystyle \varepsilon _{0}} , but cannot express ε 0 {\displaystyle \varepsilon _{0}} . There are other ordinal notations capable of capturing ordinals well past ε 0 {\displaystyle \varepsilon _{0}} , but because there are only countably many strings over any finite alphabet, for any given ordinal notation there will be ordinals below ω 1 {\displaystyle \omega _{1}} (the first uncountable ordinal) that are not expressible. Such ordinals are known as large countable ordinals.
The operations of addition, multiplication and exponentiation are all examples of primitive recursive ordinal functions, and more general primitive recursive ordinal functions can be used to describe larger ordinals.
The natural sum and natural product operations on ordinals were defined in 1906 by Gerhard Hessenberg, and are sometimes called the Hessenberg sum (or product) (Sierpinski 1958) harv error: no target: CITEREFSierpinski1958 (help). These are the same as the addition and multiplication (restricted to ordinals) of John Conway's field of surreal numbers. They have the advantage that they are associative and commutative, and natural product distributes over natural sum. The cost of making these operations commutative is that they lose the continuity in the right argument, which is a property of the ordinary sum and product. The natural sum of α and β is often denoted by α⊕β or α#β, and the natural product by α⊗β or α⨳β.
The natural operations come up in the theory of well partial orders; given two well partial orders S and T, of types (maximum linearizations) o(S) and o(T), the type of the disjoint union is o(S)⊕o(T), while the type of the direct product is o(S)⊗o(T).[3] One may take this relation as a definition of the natural operations by choosing S and T to be ordinals α and β; so α⊕β is the maximum order type of a total order extending the disjoint union (as a partial order) of α and β; while α⊗β is the maximum order type of a total order extending the direct product (as a partial order) of α and β.[4] A useful application of this is when α and β are both subsets of some larger total order; then their union has order type at most α⊕β. If they are both subsets of some ordered abelian group, then their sum has order type at most α⊗β.
We can also define the natural sum of α and β inductively (by simultaneous induction on α and β) as the smallest ordinal greater than the natural sum of α and γ for all γ < β and of γ and β for all γ < α. There is also an inductive definition of the natural product (by mutual induction), but it is somewhat tedious to write down and we shall not do so (see the article on surreal numbers for the definition in that context, which, however, uses surreal subtraction, something that obviously cannot be defined on ordinals).
The natural sum is associative and commutative. It is always greater or equal to the usual sum, but it may be strictly greater. For example, the natural sum of ω and 1 is ω+1 (the usual sum), but this is also the natural sum of 1 and ω. The natural product is associative and commutative and distributes over the natural sum. The natural product is always greater or equal to the usual product, but it may be strictly greater. For example, the natural product of ω and 2 is ω·2 (the usual product), but this is also the natural product of 2 and ω.
Yet another way to define the natural sum and product of two ordinals α and β is to use the Cantor normal form: one can find a sequence of ordinals γ1 > … > γn and two sequences (k1, …, kn) and (j1, …, jn) of natural numbers (including zero, but satisfying ki + ji > 0 for all i) such that
α = ω γ 1 ⋅ k 1 + ⋯ + ω γ n ⋅ k n {\displaystyle \alpha =\omega ^{\gamma _{1}}\cdot k_{1}+\cdots +\omega ^{\gamma _{n}}\cdot k_{n}}
β = ω γ 1 ⋅ j 1 + ⋯ + ω γ n ⋅ j n {\displaystyle \beta =\omega ^{\gamma _{1}}\cdot j_{1}+\cdots +\omega ^{\gamma _{n}}\cdot j_{n}}
and define
α # β = ω γ 1 ⋅ ( k 1 + j 1 ) + ⋯ + ω γ n ⋅ ( k n + j n ) . {\displaystyle \alpha \#\beta =\omega ^{\gamma _{1}}\cdot (k_{1}+j_{1})+\cdots +\omega ^{\gamma _{n}}\cdot (k_{n}+j_{n}).}
Under natural addition, the ordinals can be identified with the elements of the free commutative monoid generated by the gamma numbers ωα. Under natural addition and multiplication, the ordinals can be identified with the elements of the free commutative semiring generated by the delta numbers ωωα. The ordinals do not have unique factorization into primes under the natural product. While the full polynomial ring does have unique factorization, the subset of polynomials with non-negative coefficients does not: for example, if x is any delta number, then
x 5 + x 4 + x 3 + x 2 + x + 1 = ( x + 1 ) ( x 4 + x 2 + 1 ) = ( x 2 + x + 1 ) ( x 3 + 1 ) {\displaystyle x^{5}+x^{4}+x^{3}+x^{2}+x+1=(x+1)(x^{4}+x^{2}+1)=(x^{2}+x+1)(x^{3}+1)}
has two incompatible expressions as a natural product of polynomials with non-negative coefficients that cannot be decomposed further.
There are arithmetic operations on ordinals by virtue of the one-to-one correspondence between ordinals and nimbers. Three common operations on nimbers are nimber addition, nimber multiplication, and minimum excludance (mex). Nimber addition is a generalization of the bitwise exclusive or operation on natural numbers. The mex of a set of ordinals is the smallest ordinal not present in the set.
^ Feferman, S. (1964). "Some Applications of the Notions of Forcing and Generic Sets". Fundamenta Mathematicae. 56 (3): 325–345. doi:10.4064/fm-56-3-325-345.
^ Ernst Jacobsthal, Vertauschbarkeit transfiniter Ordnungszahlen, Mathematische Annalen, Bd 64 (1907), 475-488. Available here
^ D. H. J. De Jongh and R. Parikh, Well-partial orderings and hierarchies, Indag. Math. 39 (1977), 195–206. Available here
^ Philip W. Carruth, Arithmetic of ordinals with applications to the theory of ordered Abelian groups, Bull. Amer. Math. Soc. 48 (1942), 262–271. See Theorem 1. Available here
Thomas Jech (21 March 2006). Set Theory: The Third Millennium Edition, revised and expanded. Springer Science & Business Media. ISBN 978-3-540-44085-7.
Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
Sierpiński, Wacław (1958), Cardinal and ordinal numbers, Polska Akademia Nauk Monografie Matematyczne, 34, Warsaw: Państwowe Wydawnictwo Naukowe, MR 0095787
ordCalc ordinal calculator


January 6, 2016 Alamo Drafthouse, SXSW Film.
"Raiders!" has been a hit on the festival circuit with strong screenings at SXSW, DOC NYC, Fantasia International Film Festival and more.
"Raiders!: The Story of the Greatest Fan Film Ever Made" will see a limited theatrical release across North America in the summer of 2016, alongside a variety of VOD and digital platforms, with a Blu-ray/DVD release to follow later in the year.
We saw this film in March as part of South By Southwest. Here is our review.


The fate of The Celebrity Apprentice has been in limbo since late last spring and is still unclear; the show still doesn't have a debut date or a cast at a time when it's normally in production. However, at least one celebrity–Motley Crue's Vince Neil–has said he's been cast.
The Las Vegas Review-Journal reports that "Neil has been telling friends he's heading for Donald Trump's boardroom," and also notes that there are rumors Miss USA 1995 Shanna Moakler has been cast. Neil has previously appeared on The Surreal Life and Skating with the Stars.
Assuming they can scrape together a solid cast, I think the show will return, perhaps as a replacement for a cancelled show in the spring, or perhaps it will air in the summer. But if the ratings keep dropping, I wouldn't be surprised if NBC cancelled it.
However, I'd still love to see Ivanka Trump take over the franchise.


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Lincoln International Academy is a co-ed, lay school with a high academic standard and a solid training in the English language. It is characterized by being an inclusive school focused on the family. This institution is over 40 years old and throughout its history has been known to maintain the family atmosphere of its founders.
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I got mixed up in all the cougar chaos the same morning I was shot with a herring.
The herring incident happened while I was helping feed the penguins at FunJungle Wild Animal Park, early on a Saturday morning in late May, before the park had officially opened for the day. My girlfriend, Summer, was also there. I was only thirteen, and Summer was fourteen, but since Summer's father, J.J. McCracken, owned FunJungle and both my parents worked there, we often got to go behind the scenes.
First, a heat wave was frying central Texas. Normally, the temperature in late May should have merely been uncomfortably warm; instead, it was blisteringly hot. The day before, in science class, we had fried an egg on the school parking lot. Meanwhile, the penguin exhibit was chilled to twenty degrees Fahrenheit. It was the perfect way to beat the heat.
Second, we got to use a cannon.
It wasn't a real cannon. There was no gunpowder or anything like that. Instead, it was a pneumatic plastic tube created by the Zoom Corporation to move fish at high speeds.
Zoom had originally invented the cannon to help salmon get past dams in the Pacific Northwest. Salmon are born in mountain lakes, swim downstream to the ocean to mature, and then, years later, return to the exact same lakes where they were born in order to spawn. Unfortunately, dams often prevent the salmon from returning to their headwaters, and until recently, the only option had been to build expensive fish ladders, which were like giant concrete staircases the salmon could "climb" by jumping from one pool to the next. Firing the fish through a pneumatic tube over the dam was a lot cheaper—albeit somewhat ridiculous. J.J. McCracken had liked the idea, though. He had invested a good deal of money in Zoom, and while he was explaining the concept to Summer one night at dinner, she had suggested that maybe the tubes could be used for dead fish as well as live ones.
J.J. McCracken was a smart man, but he always claimed his daughter was even smarter; so when she made suggestions, he listened. (After all, Summer had come up with the whole idea for FunJungle itself when she was only seven.) Summer's logic went like this: FunJungle couldn't feed the penguins live fish, because it was hard to control parasites in a live food supply and we didn't want the penguins to get sick. So all their food was frozen and then thawed out for feeding time. In the case of the penguins, this amounted to over 700 pounds of fish a day. Normally, the keepers fed the penguins by tossing little chunks to each of them, which was very time-consuming and promoted abnormal behavior.
Which was how, two weeks later, Summer and I found ourselves in the penguin exhibit early on a Saturday morning, loading frozen herring into a fish cannon.
Getting there that early hadn't been any trouble for me; I lived in FunJungle's employee housing, which was a collection of double-wide trailer homes not far past the back fence of the park. J.J. owned ten square miles of property in the Texas Hill Country, of which FunJungle only took up a fraction. (So far, at least; J.J. was hoping to greatly expand his theme park empire in the future with additions to FunJungle and new resort hotels.) Out my door there was nothing but forest. Although I liked hiking in the woods, there was nothing else to do around there except visit FunJungle. I had my own employee pass that let me enter the park whenever I wanted, as both my parents tended to be there rather than at home.
My mother was the head primatologist at FunJungle, while Dad was the park's official photographer (though he still got away to take photos for National Geographic on occasion). Dad had accompanied me to the park that morning, but Mom had already been at work. Most animals wake up early, which means keepers have to be there early too—Mom was often on the job by five a.m.
Cindy Salerno, the head penguin keeper, lived a few trailers away from us. She was always cheerful and friendly and she baked a mean apple pie. (Although, as an occupational hazard, no matter how much she showered, she always smelled slightly like fish.) Cindy knew me well enough to trust me to help out with the cannon; she didn't know Summer that well, but J.J. had insisted Summer be there that morning because the whole thing was her idea. Cindy was excited about the fish cannon, but had felt we should give it a test run before the tourists arrived, just in case something went wrong.
This turned out to be a very shrewd idea.
At eight a.m. it was already sweltering outdoors, but inside the penguin exhibit we were dressed for winter. It was actually snowing in there. A special machine had been built for this. An enormous modified snow-cone maker shaved flakes of ice off giant cubes and blew them through vents in the ceiling. The snow then wafted down and piled up in drifts. The machine ran for a few hours every morning, generating over a ton of fresh snow a day. The penguins loved it. They were dancing in the shower of flakes, waggling their stubby wings in delight.
I enjoyed the snow quite a bit myself. I had never lived anywhere it snowed, and even though I knew this was fake, it was still fun. The only drawback was that I didn't own any serious winter clothing and had to borrow ski clothes from Summer. The parka wasn't too girly, but it still had a fringe of pink fluff around the hood. Meanwhile, we were also wearing tall rubber waders to protect ourselves from the penguins. (Penguins aren't too aggressive, but if you crowded them, they would sometimes peck your shins to make you back off.) FunJungle had only purchased waders in adult sizes—no one had ever considered that thirteen-year-olds might be wearing them—so they rode ridiculously high on our legs.
"You are a serious fashion disaster," Summer informed me as we lugged the fish cannon into the exhibit.
One of my favorite things about Summer—besides her being beautiful and smart and surprisingly down-to-earth for a really rich girl—was that she had a great sense of humor. We spent a lot of time teasing each other.
However, the penguins had no idea Summer was wielding the fish in jest. Now that it was out in the open, 416 heads swiveled toward her at once. Sensing food, the penguins began waddling toward us en masse, barking for her attention.
Summer quickly returned the fish to the cooler, setting it atop the hundreds of others stored there, then replaced the lid. The little penguin shifted its attention from Summer to the cooler, which it pecked at hungrily. "Why's he called Fifty-Six?" Summer asked.
"You didn't name them?" Summer asked, surprised.
Summer helped me pick up the cannon again and we moved it close to the edge of the water. The device wasn't too heavy, but it was unwieldy, especially the plastic tube. Dozens of penguins were now crowding around us curiously, and we had to be careful not to knock any of them over.
The exhibit was similar to virtually every other penguin exhibit on earth: The "land" portion of it was in the back, while the watery portion was up against the viewing glass, so that guests could watch the penguins swim. Swimming penguins were quite fascinating; they moved with amazing speed and grace. Meanwhile, penguins on land tended to be less interesting. Usually, they merely stood around in clumps, occasionally waddling from place to place or firing projectiles of poop into the water.
So we did. The cannon was a relatively simple device. The bulk of it was a small blower that created a differential in pressure in the plastic tube, sucking the fish through it. If we cranked the blower to the highest level, it could actually move the fish at twenty miles an hour, but for the time being, Sanjay had suggested we set it much lower. I flipped the power on and the cannon came to life with a low hum.
It was quiet enough that only the closest penguins took notice. The rest of them remained standing in their little clumps and squawking at one another.
"More than the capelin?" I asked.
Summer and I grabbed the tube and directed the end of it at the aquatic portion of the exhibit.
Now that Cindy had a herring in her hands, the penguins' attention returned to us again. They all began waddling our way once more.
A small metal crate held the blower and the loading end of the pneumatic tube. Cindy placed the herring at the end, and it was instantly sucked inside.
The fish moved through the tube quickly, a small bulge rippling down the length of it. It looked like a python swallowing a large meal, sped up a thousand times. The fish shot past our hands and fired out the other end, plunging into the water.
Even though many of the penguins had been bred in captivity, the motion of the fish seemed to stimulate their innate instincts. A dozen of the penguins closest to the water's edge dove right in after the fish. A battle quickly ensued beneath the surface for it. A few of the smaller penguins got a morsel, but a big emperor labeled "99" won out and gulped the herring down.
In the viewing area, Sanjay Budhiraja gave a whoop of joy loud enough for us to hear him through the glass.
"Let's do some more!" Summer said.
Cindy grabbed another herring and fed it into the cannon. This time, I aimed the tube at a spot away from Ninety-Nine, since he'd already eaten. The second fish fired into the water. The penguins that were already swimming darted that way, while a whole new clump of penguins on land plunged in after it.
Cindy scooped up some capelin and we shot these into the water as well. Now all the penguins in the exhibit sensed the presence of food and began migrating toward the water's edge. Cindy was thrilled, Sanjay was still whooping, and Summer and I were cheering as we watched the penguins race around for the fish. The penguins either didn't know the fish weren't alive, or they didn't care. Whichever the case, it was great fun to watch, and we were all enjoying how well it was all working.
At which point everything took a turn for the worse.
The crowd of penguins along the water's edge was starting to get very big. They were so interested in the fish they didn't seem to care about us at all. They were congregating all around us. It was a little unsettling, especially when the emperors, which could be over four feet tall, shoved up against us. A particularly large one—335—actually tried to nudge Summer out of its way. Summer stumbled backward and stepped right on a poor little gentoo that was scurrying between her legs. The gentoo squawked in pain and Summer dropped the pneumatic tube in surprise.
The tube now snapped from my hands as well, just as a large herring was moving through it. Without anyone holding it, the tube whipped around on the ground, writhing like an angry snake among the penguins.
This now startled the birds, who stampeded away from it. Penguins aren't very adept on land, though. A stampede of them wasn't threatening so much as slow and clumsy. Dozens tripped and fell on their faces. Their fellow penguins either ran right over them or tripped over their prone bodies. Many who were waiting by the water's edge were shoved off and belly flopped into the pool.
The whole thing might have been comical if we had been watching it, rather than caught in the middle of it. A surge of panicked penguins knocked Cindy off her feet. She fell backward onto the fish cooler, which toppled, spilling its entire contents into the loading area for the fish cannon. The tube promptly began sucking up the fish and firing them out the other end.
Which might not have been a problem if a passing macaroni penguin hadn't bumped the dial for the blower, cranking it to top speed.
Now the fish came blasting out of the tube at twenty miles an hour in rapid succession. The fish cannon had become more of a machine gun. And without anyone holding it, the tube kept writhing around wildly, firing piscine projectiles in random directions. Frozen herring ricocheted off the fake icebergs and plowed into snowdrifts. A half dozen capelin slammed into the viewing glass and exploded on impact, leaving splatters of fish guts. A few unfortunate penguins got caught in the crossfire and were clocked by flying fish. None were seriously hurt, but each strike resulted in a cacophony of pained squawks and a slight burst of feathers.
Now the penguins grew even more panicked. They fled in every direction, unsure what was safe, a surging, chaotic sea of black and white. With their little tuxedoed bodies, it looked like someone had yelled "Fire!" at the Oscars.
Summer and I were scrambling about, dodging penguins as we tried to grab the tube, but as more and more fish shot through it at greater and greater speeds, it was thrashing about more wildly than ever. With one spasmodic jolt, it clotheslined Chicken à la King and sent the poor bird cartwheeling into the water.
Meanwhile, Cindy Salerno was trying to turn off the cannon, but a horde of penguins was crowded around the controls, gulping down the fish that hadn't been slurped up by the hose. In the midst of the crowd, one of the smallest penguins—a juvenile Adélie—got too close to the intake. With a sudden gasp of air—and a startled squawk—it was sucked inside. Since the little guy was equally as streamlined as the fish, he rocketed through the tube and fired out the other end, becoming the first penguin in history to fly.
Unfortunately, I was right in its path. The Adélie bore down on me, beak first, screaming in penguin terror and flapping its wings madly, as if desperately willing them to work. I dove for cover into a snowdrift and the penguin sailed over my head, soared the length of the exhibit, and plopped safely into the water.
A flock of frightened penguins quickly stampeded over my back. A few stepped right on my head, smushing my face into the snow. I flailed my arms, scattering them, and staggered back to my feet, spitting out ice flakes. . . .
Which was when the herring hit me.
It was an exceptionally large herring, and it caught me right in the chest.
Summer had finally managed to grab the tube and steady it, but not before realizing it was pointing directly at me. The herring hit me at full speed, hard enough to knock the wind out of me and throw me off balance. I stumbled backward, tripped over a chinstrap penguin—and fell into the water.
Penguins prefer their water very cold. At some places in the Antarctic, due to some interesting physics about ice and salt water, the ocean can actually be colder than freezing temperatures. The water in the exhibit wasn't quite that frigid, but it was close enough. It felt like my whole body had been slapped. Every one of my muscles tensed at once. And if the cold wasn't bad enough, I was also in a maelstrom of agitated penguins and half-eaten fish.
Many of the fish had been too big to eat in one gulp, so the penguins had torn them to bits instead, leaving clouds of white flesh and fish guts in the water. Penguins were darting through it all, gulping down what they could, moving far faster in the water than on land. It was like being surrounded by a swarm of fighter pilots, who dipped and whirled and corkscrewed around me.
My hip-length boots immediately filled with water and dragged me down, while Summer's ski jacket became saturated and heavy as an anchor. I shrugged everything off as fast as I could. Luckily, the water wasn't deep, so I quickly broke back through the surface.
Cindy was there, waiting for me, having finally turned off the fish cannon and waded through the penguins to my rescue. She grabbed my arms, hauled me back onto land, and wrapped her own parka around me to keep me from freezing solid. "Let's get you warm," she said, and hustled me toward the door.
Now that the cannon was off and the penguins had devoured all the loose fish, the birds had calmed down. The panic was over and they were now milling about, preening themselves as if nothing had happened.
I wasn't doing as well. The chill of the water had already sunk into my bones. My legs were trembling so badly I needed both Summer and Cindy to steady me as we hurried through the exhibit.
The door out of the exhibit was concealed behind a fake glacier. We passed through it into the keeper's area. This wasn't much warmer, but after the arctic temperatures I had just been subjected to, it felt like I had suddenly gone to the tropics.
I suddenly realized there was another person in the room. She was a young adult—barely past college age, if that—and short but extremely fit. Her hair was cropped short, almost in a crew cut, and her nose was pierced. She wore a T-shirt and cargo shorts. There was something strangely familiar about her face, though I couldn't tell what. She was sitting in Cindy's chair, quietly observing everything.
Cindy seemed surprised by her presence too; she didn't appear to know the woman. "Are you with Zoom too?" Cindy asked.
"It's a company that makes fish cannons," Sanjay said, as if that would make sense to anyone.
I suddenly understood why she looked so familiar; she looked like Doc, the head vet at FunJungle. Not exactly like him, which would have been weird. But she was definitely the younger and more feminine version of him.
I had heard of Lily before, but never met her. Or even seen a photo of her.
"What are you doing in my office?" Cindy asked.


text:
The 2011 AFC Beach Soccer Championship was a continental beach soccer tournament, which took place from February 27 to March 4, 2011, in Muscat, Oman, seeing the tournament leave Dubai for the first time. Beach Soccer Worldwide and FIFA decided to move the location of the championship to the Al-Musannah Sports City complex in Muscat, due to the success of the 2010 Asian Beach Games held there.

The two finalists and the third place play-off winner earned qualification to represent Asia at the 2011 FIFA Beach Soccer World Cup in Ravenna, Italy being Japan, who won the championship for a second consecutive time, Oman who finished in second place, qualifying to the world cup for the first time and Iran who beat the United Arab Emirates in the third place play off, to qualify by finishing in third place for the fourth time.

Participating teams
A tournament record 11 teams, an increase of four teams following the seven that participated in the 2009 qualifiers confirmed their participation in the competition, with many newcomers, showing the ever growing popularity of the sport.

Originally, 15 teams were going to take part, which included Afghanistan, Palestine, Qatar and Yemen. However, the four countries pulled out of the tournament because of an undisclosed reason.

Group stage
The draw to divide the eleven teams into the following three groups was conducted on 24 February 2011. The group stage commenced on 27 February 2011 and consisted of each nation playing each other once in a single round-robin format. To qualify for the quarter finals, a team must have finished in the top two of each group or have one of the two best records of the third placed group finishers.

All kickoff times are listed as Omani local time (UTC+4).

Group A

Group B

Group C

Third placed teams
Because some groups contain four teams and some three, matches against the fourth-placed team in each group are not included in this ranking.

Knockout stage

Quarter finals

Semi finals

Third place play off

Final

Winners

Awards

Top scorers

8 goals
 Takeshi Kawaharazuka
 Ishaq Al-Mas
7 goals
 Hussein Jabbar
 Shusei Yamauchi
 Ali Hassan Karim
6 goals
 Yahya Al-Araimi
 Jamoliddin Sharipov
5 goals
 Rashed Jamal
 Mohammad Ahmadzadeh
 Jalal Al-Sinani
 Rashid Ahmed Hassan
4 goals
 Han Xue
 Mehdi Hassani
 Moslem Mesigar
 Khaild Al-Zaabi
 Adel Ranjabar
3 goals
 Gui Qixuan
 Farid Boloukbashi
 Ali Naderi

3 goals (cont.)
 Omar Al-Jaser
 Abdulwahhab Al-Safi
 Youssef Al-Saqer
 Yaqoob Al-Alawi
2 goals
 Mohamed Darwish
 Ebrahim Hasan
 Yaqoob Nesuf
 Talal Yousef
 Qiu Hao
 Hassan Abdollahi
 Farough Dara
 Sadeq Jabr
 Takashi Arakaki
 Hirofumi Oda
 Teruki Tabata
 Haitham Al-Araimi
 Mohammad Nassif
 Javlon Anorov
 Fakhriddin Samegov
 Abdusattor Sattorov
1 goal
 Yilihanmu Aihaiti
 Gede Darmasuta
 Dewa Kadek Dwipayudha

1 goal (cont.)
 Nyoman Jumada
 Komang Kariana
 Wayan Metrajaya
 Juli Parimawan
 Moshtak Kadhim
 Kunihiro Wakabayashi
 Abdulaziz Al-Eid
 Hani Al-Dhabit
 Omar Akil
 Omar Al-Najjar
 Alaa Arnaout
 Ahmad Shatta
 Humaid Al-Balooshi
 Rami Al-Mesaabi
 Walid Mohammed
 Qambar Sadeqi
 Furkat Azizov
 Nodir Elibaev
Own goal
 Ishaq Al-Mas (for Kuwait)

Final standings

References
Day 1 Results
Day 2 Results
Day 3 Results and quarter final draw
Day 4 Quarter final Results

Beach Soccer Championship
2011
Qualification Afc
2011
2011 in beach soccer
Beach

redpajama_set_name: RedPajamaWikipedia


text:
>> Tuesday, August 10, 2021
Football super star Lionel Messi joining the Qatar owned French club Paris Saint Germain, PSG. Rumours for the last one week after his surprise exit from his club Barcelona.
It's all been confirmed. Lionel Messi is going to be a PSG player," football pundit Guillem Balague told BBC Radio 5 Live. "That's it, it's done. It's happened in the last few hours," Balague added.
Fans eagerly waiting for Messi at Paris
Messi heading to airport en route to Paris


text:
Home/American Shipper/DOT cross-border program faces delay, changes
DOT cross-border program faces delay, changes
American Shipper StaffSunday, April 29, 2007Last Updated: Saturday, March 2, 2019
The U.S. Department of Transportation responded to vocal criticism and lawsuits Monday by changing two key aspects of a proposed demonstration project to allow Mexican-domiciled trucks to carry cargo deep into the United States.
Congressional leaders, activists and labor unions have been critical of the lack of public notice and input about the cross-border program since it was announced by the DOT early this year.
In response, the DOT said Monday it would publish official details of the program in the Federal Register May 1 and accept comment for 30 days.
In recent weeks, the Teamsters Union, the Sierra Club, Public Citizen and the Environmental Law Foundation have all filed suit over the public transparency concerns.
Congressional leaders in Congress said Transportation Secretary Mary E. Peters, in making the changes, had effectively chosen to postpone implementation of the project, which was scheduled to start any day.
"This delay will provide Congress with an opportunity to evaluate the proposal and its impacts on safety and on commerce," said Rep. James L. Oberstar, D-Minn. Oberstar, head of a key transportation committee, has been critical of the program since its announcement by the administration.
"I continue to be highly concerned about the preparedness of Mexican trucking companies and their drivers to abide by the same strict standards that apply to U.S. motor carrier operations," he said.
In a second change, Peters said in a statement that U.S. trucks would be able to operate in Mexico from the start of the program. The original schedule for the plan, which had also faced considerable criticism from labor and trucking groups, called for U.S. trucks to have access to Mexico six months after the program began letting the Mexico-domiciled trucks enter the U.S.
Sen. Dianne Feinstein, D-Calif., told the Associated Press that the earlier entry time for U.S. trucks was "good news" that will "restore fairness to this important program."
When the project was first announced in February, Peters said Mexican trucks could enter the United States by late April. While the department has not said when the program would begin, it will likely be sometime after the 30-day comment period announced Monday.
The pilot program will allow 100 Mexican truck companies to operate 1,000 Mexico-domiciled trucks beyond a current 20-mile zone just inside the U.S.-Mexican border.


text:
Q: Mutual Independence Definition Clarification Let $Y_1, Y_2, ..., Y_n$ be iid random variables and $B_1, B_2, ..., B_n$ be Borel sets. It follows that
$P(\bigcap_{i=1}^{n} (Y_i \in B_i)) = \Pi_{i=1}^{n} P(Y_i \in B_i)$...I think?
If so, does the converse hold true? My Stochastic Calculus professor says it does (or maybe misinterpreted him somehow?), but I was under the impression that independence of the n random variables was equivalent to saying for any indices $i_1, i_2, ..., i_k$ $P(\bigcap_{j=i_1}^{i_k} (Y_j \in B_j)) = \Pi_{j=i_1}^{i_k} P(Y_j \in B_j)$.
So, if the RVs are independent, then we can choose $i_j=j$ and k=n to get $P(\bigcap_{i=1}^{n} (Y_i \in B_i)) = \Pi_{i=1}^{n} P(Y_i \in B_i)$, but given $P(\bigcap_{i=1}^{n} (Y_i \in B_i)) = \Pi_{i=1}^{n} P(Y_i \in B_i)$, I don't know how to conclude that for any indices $i_1, i_2, ..., i_n$ $P(\bigcap_{j=i_1}^{i_k} (Y_j \in B_j)) = \Pi_{j=i_1}^{i_k} P(Y_j \in B_j)$, if that's even the right definition.
p.17 here seems to suggest otherwise. idk
Help please?
Also this:

or

So, this answer is to use the Omega part to establish pairwise independence and ultimately conclude independence. Without that assumption, we cannot conclude independence. Is that right? Why does that not contradict the definition of independence: $P(\bigcap_{i=1}^{n} (Y_i \in B_i)) = \Pi_{i=1}^{n} P(Y_i \in B_i)$ ?

A:
I was [under][2] the [impression][3] that independence of the n random variables was equivalent to saying for any indices $i_1, i_2, ..., i_n$ $P(\bigcap_{j=i_1}^{i_n} (Y_j \in B_j)) = \Pi_{j=i_1}^{i_n} P(Y_j \in B_j)$.

You misread: independence of $n$ random variables $(Y_1,\ldots,Y_n)$ is equivalent to the following condition:

(C) For every distinct indices $i_1, i_2, ..., i_k$ and every $B_j$, $P(\bigcap\limits_{j=i_1}^{i_k} (Y_j \in B_j)) = \prod\limits_{j=i_1}^{i_k} P(Y_j \in B_j)$.

Indeed, choosing $k=n$ and $i_j=j$, (C) implies condition (C'):

(C') For every $B_j$, $P(\bigcap\limits_{i=1}^{n} (Y_i \in B_i)) = \prod\limits_{i=1}^{n} P(Y_i \in B_i)$.

In the other direction, if (C') holds, then, for every distinct indices $i_1, i_2, ..., i_k$ and every $B_j$, one can complete the collection of events by $(Y_s\in\mathbb R)$ for the $n-k$ missing indices $s$, then (C) follows.

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Leikanger Church
Leikanger, Norway
Leikanger Church was originally built of stone in ca. 1250. Two towers and the porch were added in the 1600s. In 1872 the interior was replaced almost completely, the porch was demolished and tower replaced with a new one. The pulpit and altar date from the early 1600s.
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Kyrkjevegen 4, Leikanger, Norway
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Palazzo Colonna
The Palazzo Colonna is a palatial block of buildings built in part over ruins of an old Roman Serapeum, and has belonged to the prestigious Colonna family for over twenty generations.
The first part of the palace dates from the 13th century, and tradition holds that the building hosted Dante in his visit to Rome. The first documentary mention notes that the property hosted Cardinal Giovanni and Giacomo Colonna in the 13th century. It was also home to Cardinal Oddone Colonna before he ascended to the papacy as Martin V (1417–1431).
With his passing, the palace was sacked during feuds, and the main property passed into the hands of the Della Rovere family. It returned to the Colonna family when Marcantonio I Colonna married Lucrezia Gara Franciotti Della Rovere, the niece of pope Julius II. The Colonna"s alliance to the Habsburg power, likely protected the palace from looting during the Sack of Rome (1527).
Starting with Filippo Colonna (1578–1639) many changes have refurbished and create a unitary complex around a central garden. Architects including Girolamo Rainaldi and Paolo Marucelli labored on specific projects. Only in the 17th and 18th centuries were the main facades completed. Much of this design was completed by Antonio del Grande (including the grand gallery), and Girolamo Fontana (decoration of gallery). In the 18th century, the long low facade designed by Nicola Michetti with later additions by Paolo Posi with taller corner blocks (facing Piazza Apostoli) was constructed recalls earlier structures resembling a fortification.
The main gallery (completed 1703) and the masterful Colonna art collection was acquired after 1650 by both the cardinal Girolamo I Colonna and his nephew the Connestabile Lorenzo Onofrio Colonna and includes works by Lorenzo Monaco, Domenico Ghirlandaio, Palma the Elder, Salviati, Bronzino, Tintoretto, Pietro da Cortona, Annibale Carracci (painting of The Beaneater), Guercino, Francesco Albani, Muziano and Guido Reni. Ceiling frescoes by Filippo Gherardi, Giovanni Coli, Sebastiano Ricci, and Giuseppe Bartolomeo Chiari celebrate the role of Marcantonio II Colonna in the battle of Lepanto (1571). The gallery is open to the public on Saturday mornings.
The older wing of the complex known as the Princess Isabelle"s apartments, but once housing Martin V"s library and palace, contains frescoes by Pinturicchio, Antonio Tempesta, Crescenzio Onofri, Giacinto Gimignani, and Carlo Cesi. It contains a collection of landscapes and genre scenes by painters like Gaspard Dughet, Caspar Van Wittel (Vanvitelli), and Jan Brueghel the Elder.
Along with the possessions of the Doria-Pamphilij and Pallavacini-Rospigliosi families, this is one of the largest private art collections in Rome.
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AWOL-72 (2015) with Luke Goss Movie DVD
Black Gold Movie on DVD
Drug Wars: Silver or Lead (2008) Movie DVD
Far Out Isn't Far Enough: The Tomi Ungerer Story (2012) on DVD
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Home / Uncategorized / TV Series and Movies / Making a Murderer Complete Season 2 (2018)
Making a Murderer Complete Season 2 (2018)
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Feel free any question you might have.
If there are other titles you are looking for don't hesitate to ask.
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Strictly Come Dancing Complete Season 12 (2014)
The Choir 1995 starring David Warner
The Great British Bake Off Season 5 (2014)
The Haves and the Have Nots 2018 (aka Season 5 Part 2)
La Mante 2017 (The Mantis) with English Subtitles
Insured for up to $10000 by Sectigo
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Pod prikritie Seasons 3, 4 and 5 Complete with English Subtitles
Beat the Chasers Season 1 (2020) on DVD
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The Repair Shop Seasons 1, 2, 3, 4, 5 and 6 on DVD
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text:
var Faker = require('../lib');
var faker = new Faker({ locale: 'it', localeFallback: 'en' });
faker.locales['it'] = require('../lib/locales/it');
faker.locales['en'] = require('../lib/locales/en');
module['exports'] = faker;

redpajama_set_name: RedPajamaGithub


text:
Capim Branco es un municipio brasileño del estado de Minas Gerais. Su población estimada en 2018 es de 9 679 habitantes, según el Instituto Brasileño de Geografía y Estadística (IBGE). Integra la región metropolitana de Belo Horizonte.

Historia y toponimia
El actual municipio de Capim Branco se originó en 1890 como distrito del municipio de Pedro Leopoldo, del cual formaba parte hasta 1943. A partir de ese año integró el municipio de Matozinhos, y en 1953 obtuvo la autonomía. El topónimo se debe al hecho de medrar en sus tierras, en gran cantidad, cierta especie de pasto que tiene el color blanco.

Clima
Según la clasificación climática de Köppen, el municipio se encuentra en el clima tropical de altitud Cwa.

Referencias

Enlaces externos

 Sitio web oficial .

Localidades de Minas Gerais


This topic contains 4 replies, has 2 voices, and was last updated by largebla 6 years, 6 months ago.
Hello everyone, Is there a link on here about understanding Bloodlines.
How can I research this more on the LBH. I need to know how to look at creating our own bloodline. I hope I'm phrasing this right.
You cannot create your own bloodline. You can create your own herd name like ours is Cornish. The bloodlines stay the same from generation to generation. The male (boar) piglets ALWAYS take the name of the sire & the female (gilt) piglets ALWAYS take the name of the sow.
SO for example if you mate Cornish Majestic 11/5 1219 (Pet Name Mick) to RF MATILDA 2/3 *0347 (Pet Name Molly) the boar piglets born will be Majestics & the the gilts will be Matildas. This actually makes reading & understanding the registration papers / genealogies easy. Believe you me, some pedigrees out there for other animals are so confusing it, it isn't even funny!
Good Morning Liz, I see now, and it makes good sense that your not creating a new bloodline ( What a Rookie I, am, lol) So I know get what folks are doing, breeding what they think are great genetics and putting their name on them.
Thanks Liz, You are the Master, and I am the Student.
Hope Richard is feeling better.
Did you get a chance to watch the Video I sent you?
LOL! No Master here – still learning but just trying to help folks learn & not make the same mistakes that we have made.
Your help, is Very much Appreciated.


text:
Q: How to run a standalone cli script using a nextjs project? I have an working nextjs project.
I need to create a script that I can run from the CLI that uses some of the CRUD libraries I've written for the next project.
**/scripts/backup-assets.js **
import {getAllProjectsData} from '../lib/api/projects'

async function main() {
  const allProjectsData = await getAllProjectsData()
  console.info({allProjectsData})
}

main()

I'm getting this error:
$ node scripts/backup-assets.js
(node:9736) Warning: To load an ES module, set "type": "module" in the package.json or use the .mjs extension.
(Use `node --trace-warnings ...` to show where the warning was created)
C:\Users\...\scripts\backup-assets.js:1
import {getAllProjectsData} from '../lib/api/projects'
^^^^^^

SyntaxError: Cannot use import statement outside a module
    at Object.compileFunction (node:vm:352:18)
    at wrapSafe (node:internal/modules/cjs/loader:1033:15)
    at Module._compile (node:internal/modules/cjs/loader:1069:27)
    at Object.Module._extensions..js (node:internal/modules/cjs/loader:1159:10)
    at Module.load (node:internal/modules/cjs/loader:981:32)
    at Function.Module._load (node:internal/modules/cjs/loader:822:12)
    at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:77:12)
    at node:internal/main/run_main_module:17:47

A: Use TSX. It runs out of the box with NextJS, with both JavaScript and TypeScript.
Install it:
npm install --save-dev tsx

Declare your script. In package.json:
...
"scripts": {
  ...
  "my-js-script": "tsx scripts/my-script.js",
  "my-ts-script": "tsx scripts/my-script.ts"
},

Run it:
npm run my-js-script
npm run my-ts-script

A: I haven't found a way to do this but here's a workaround:

*

*Create an API route called /api/scripts which takes a task param.

*Add a check to prevent it running in production (or authentication if you'd like to run your scripts in production context).
export default async function handler(req, res) {

  if (process.env.NODE_ENV === 'production') return res.status(404).end();

  switch (req.query.task) {
    case 'backup-assets':
      // code
    case 'migrate-database':
      // code

*Launch in dev or production mode as appropriate:

npm run dev

*Run the desired script from the command line with curl:

curl http://localhost:3000/api/scripts?task=backup-assets

or just visit that URL in the browser.

A: Here's how I managed to set up standalone deployment/maintenance scripts in my NextJS project.
First, install ts-node (CLI runner) and tsconfig-paths as local DEV dependencies. The last one is a ts-node plugin that loads path aliases from tsconfig.json. If you're using plain JavaScript or don't use path aliases, you can skip it.
npm i -D ts-node tsconfig-paths

Then add a script entry to your package.json:
...
"scripts": {
  ...
  "recreate-image-info": "ts-node -O '{\"module\": \"commonjs\"}' -r tsconfig-paths/register deploy/recreate-image-info.ts"
},

The -r flag with its parameter enables the tsconfig-paths plugin. The -O flag specifies compiler options.
Then, to run the script via CLI:
npm run recreate-image-info


Did you know that with a Ghanaian international passport, you can visit over 63 countries around the world VISA FREE or on VISA ON ARRIVAL basis?
According to the ECOWAS arrangement, a west African can reside for up to 90 days in any West African country without a visa. If you have a current passport, you'll notice it also serves as an ECOWAS travel permit. So, if you have been wondering the places you can travel to with a passport of Ghana and without a visa, here's your treat! There are lots and lots of them.
Several Commonwealth countries would allow you visit their country without any visa. Yes, without applying or obtaining a visa, it's that simple. However, you should be aware that there's a stipulated length of time for which you can stay in these countries without a visa. When this period elapses, you'll be required to obtain a visa. Some countries listed here would grant you visa upon arrival for a small fee.
Philippines: 21 Days – For more than 21 days and up to 59 days, a visa would be issued on arrival.
Below are more countries that offer fewer traveling restrictions to Ghanaians. All the listed countries would allow you to travel without a visa. However, most of the countries in this section would preferably, issue a visa on arrival. This removes the hassles and delays involved in securing a visa before traveling as the visas are almost always, granted once you have a genuine reason.
Nonetheless, it is imperative to check with the country's immigration very close to the time of your travel as visa policies change frequently.
Egypt: 15 Days – Valid for South Sinai resorts only.
Iraq: Only available at Erbil international airport.
Zimbabwe: 3 Months – 3 months for holiday, 30 days for business.


text:
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Brooklyn Fire Proof Stages, home to the Nut Roaster Stages and the Seltzer Rooms Studios, is a Qualified Production Facility (QPF), participating in the New York State Film Production Tax Credit Program. Over 17,000 square feet of full-service sound stages, our five studios include a 7,200 sq. ft. stage, complete with 35′ wide cyclorama, two dedicated green screen studios, a 1,700 sq. ft. drive-on stage and two 1,000 square foot interview rooms. Amenities include ample support spaces, lighting grids, heavy power, parking, production services, and exterior locations, including a 16,000 sq. ft. roof with views of the Manhattan skyline. BFP is also a grip and electric equipment rental house, offering expendables, camera support, and theatrical fixtures. BFP is the movie production and photography epicenter of the Bushwick art community. Located in the center of the Brooklyn Fire Proof arts complex, a series of seven interconnected and thriving creative businesses, in the industrial Bushwick/East Williamsburg section of Brooklyn. It's the perfect spot to shoot your next feature film, television series, commercial, music video, photo shoot, or multimedia project.
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Shop for custom jewellery Sydney area at best price at The Dymocks Building. Your destination for unique custom made jewellery for your special occasions. We have stores of Jewellers specialise in custom made fine jewellery, including stunning wedding and engagement rings, necklaces, bracelets and more. Shop now for best shopping experience.
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Christian Chat | Testimony Tree
Brand or Product » Website
Page generated in 0.3716 seconds with 20 queries and GZIP enabled on 172.31.7.56.


text:
Fairview, Nevada
Fairview Details
Status: Ghost Town
Elevation: 4,669 Feet ( 1,423 meters)
Gallery: View 5 Fairview, Nevada Photos
County: View Churchill County Mines
Date Settled: 1906
Current Population: None
Peak Population: Around 2,000
Primary Mineral: Silver
Map: View on Google Maps
Fairview History
Fariview had a post office from 1906 to 1919.
Discoveries of silver ore in 1905 and early 1906 resulted in the founding of Fairview by summer of 1906. There was initially quite an excitement at the new town, and thousands of hopeful miners, prospectors, and businessmen arrived during that first summer.
Upper Fairview, Nevada ca. 1913
By early 1907 Fairview had an estimated 2,000 residents, and over 20 saloons to keep them busy when not engaged in mining. Fairview quickly established the necessities of a mining boom town, including hotels, banks, and a newspaper.
Ore is stacked at a Fairview mine waiting to be shipped by wagon
By 1908 much of the excitement had worn off and most of the people moved on to other districts. The mines at Fairview continued to produce however, and a town of at least a couple hundred endured for another decade.
Wagon trains prepare to haul ore to distant rail terminals
The majority of early mining at Fairview was conducted by leasers and the district lacked a stamp mill and ore had to be hauled by wagon to distant rail terminals. The result was only high-grade ore was mined during the first few years. This changed in 1911 with the Nevadahills Mining Co. built a large mill that could profitably treat lower-grade ores.
Nevadahills mll at Fairview, Nevada
With the establishment of Nevadahills as the major operator in the district, the town of Fairview moved up the hill to be closer to the mine. By 1914 the original townsite had been abandoned in favor of upper Fairview.
Nevadahills contiued operations until 1917. After that leasers worked the mines for a number of years but on a small scale. Fairview faded and and the post office closed in 1919. Today Fairview is a ghost town.
We would like to thank Burgex Inc. for being a valued partner and supporting our work.


text:
package org.unidal.cat.core.document;

import org.junit.runner.RunWith;
import org.junit.runners.Suite;
import org.junit.runners.Suite.SuiteClasses;
import org.unidal.cat.core.document.markdown.MarkdownTest;

@RunWith(Suite.class)
@SuiteClasses({

MarkdownTest.class,

})
public class AllTests {

}


text:
About InPublishing | Contact Us | Advertise
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FEATURE 27/07/2017
Well, who saw that coming!
In terms of shock value, the 2017 election result was even more unexpected than last year's Brexit vote and Trump win! No one saw it coming. Liz Gerard looks at how our national press covered the election and what effect, if any, they had on the outcome.
By Liz Gerard
Within an hour of the EU referendum result, Sun editor Tony Gallagher was crowing to the Guardian: "So much for the waning power of the print media." But that chicken came home to roost last month when, for all their efforts, Brexiteer papers were unable to deliver the expected Commons majority for Theresa May.
It had all seemed straightforward when she made her Erdogan-style pitch for an elected dictatorship in April. Conveniently forgetting the 16m who voted Remain and 12m who abstained, May had claimed that "the whole country" was behind Brexit, and MPs should be too. With the Tories 24 points ahead in the polls and Jeremy Corbyn's approval rating at -42, the result was a foregone conclusion.
"Blue Murder" cried the Sun, "PM's snap poll will kill off Labour". The Times was equally sure, announcing "May heads for landslide", while the Mail relished the opportunity to "Crush the saboteurs" – ie, any MP with the temerity to question the Dacre-approved interpretation of the Brexit vote.
With the actual vote a mere formality, no one seemed to take the election seriously: Labour's opening gambit was to offer four more bank holidays, the Tories' to bring back fox-hunting. These were clearly issues at the front of every voter's mind.
The biggest-selling papers were equally relaxed. After an initial flurry of favourable Theresa headlines, the campaign barely made the front pages. The Mail was more interested in wayward doctors, the Express in dementia and diabetes, the Sun in footballers and giant bunnies dying on transatlantic flights. The Mirror had a couple of pops at May, but was more focused on dead babies. And they all dropped everything for the latest Madeleine "breakthrough" that magically surfaced in time for the tenth anniversary of her disappearance, but which has yet to yield any further results.
The local elections reinforced the air of confidence. "Theresa on the march" proclaimed the Mail, there was a "Great Tory rebirth" in the Telegraph, and The Times repeated its earlier prediction: "May on course for landslide".
Who backed who
The Times did at least attempt to report the campaign with some level of impartiality; only the i was more even-handed. These two and the Guardian also acknowledged the existence of the LibDems and the Greens, parties almost entirely ignored by the Telegraph and the tabloids. They preferred to treat the campaign as a presidential run-off.
The Sun, Mail, Mirror and Express were all shamelessly partisan, mixing up news, speculation and comment. There were sins of commission – Labour tax "policies" that were pure invention - and of omission – having "crusaded" for years against hospital parking charges, the Express declined to report that Labour intended to abolish them.
Diane Abbott's car-crash interview on LBC was given double-page spreads all over the place, as was Jeremy Corbyn's struggle on Woman's Hour. But when the chancellor was £20bn out on the cost of HS2 in the Today programme's prime interview slot and Michael Gove got in a muddle over NHS charges for migrants, barely a sentence appeared in print.
May's mistakes were glossed over or ignored, the "dementia tax" being the prime example. On May 1, the Mail led with a story headlined "Elderly care tax breaks lifeline". It couldn't have been more wrong. When the manifesto spelt out the Conservative social care funding policy a couple of weeks later, it was still positive: "You won't have to sell home to pay for your care". And when that didn't go down too well, it came back with, "At last, a PM not afraid to be honest with you".
The Express, the paper most concerned with such issues, trumpeted "May's plan for a fairer Britain" and responded to the ensuing outcry with a small inside story headlined "May blasts critics". The next day, in what was possibly the most cynical piece of "science" reporting, it splashed on "Alzheimer's cure hope".
The Tory tabloid trio learnt two lessons from the success of their Brexit campaigns:
• one-sided reporting, bordering on propaganda, works
• when the going gets tough, play the immigration card.
But they were too blinkered to learn from the failure of David Cameron's Remain strategy:
• voters don't like Prime Ministers taking them for granted
• a negative message brings negative results.
Project Fear – Mark 2
Having mocked "Project Fear" last year, the three tabs and the Telegraph went full throttle on Jezza Jeopardy. And when the poll gaps narrowed as Corbyn was seen openly engaging with voters in their thousands while May huddled in closed encounters with handfuls of supporters, the attacks became more fevered:
Corbyn's manifesto to take Britain back to the 1970s, Corbyn to bankrupt Britain, Corbyn to open Britain's doors even wider, Corbyn's sly death tax, Corbyn's kick in the teeth for IRA victims, Corbyn doesn't believe in Britain, Corbyn's plot to bring in migrant workers, Corbyn puts UK at terror risk, Corbyn's la-la land, Corbyn plans to put 1m in top tax bracket, Corbyn accused of honouring Palestinian terror chief, Corbyn in IRA furore…
And these were just some of the front-page lead headlines; inside there was page after page of this stuff. They just about avoided blaming him for the terrorist attacks in Manchester and London, but found him culpable for atrocities all over the world over the previous 30 years. This culminated on election eve in thirteen straight pages from the Mail, starting with a splash that denounced Corbyn, John McDonnell and Diane Abbott as "Apologists for terror".
But it didn't work.
May won the election, but she lost her majority. Her party is now behind in the polls for the first time and her own approval ratings are deep in negative territory, while Corbyn's on the up.
Print vs Social Media
So, does this mean that the power of the printed media is, after all, waning? Possibly.
It is certainly true that the power of social media is waxing.
Post-election polls by YouGov and Lord Ashcroft – both with huge samples in the tens of thousands – appear to confirm that young voters were a big factor in Labour's resurgence. It is well-documented that younger generations would rather get information from the web than buy a newspaper, and Labour was more astute than the Conservatives in using Twitter, Facebook and other social sites to bypass editorial filters and get its message direct to voters.
But it would be a mistake to think that accessing material online is the sole province of the young. The referendum has left millions of people feeling politically homeless – and made them more willing to seek out and spread news reports and commentary from a wide range of sources. In this world, a blog post or story from The Canary or Guido Fawkes is as likely to find an audience as a BBC report or an FT op-ed column.
The interesting thing is whose work is shared and how. As the Mail noted in its rant about "fake news and the fascist left" last month, the Guardian may have a small circulation, but its material is shared on the internet: Owen Jones and Polly Toynbee reach thousands who don't take their paper. But web users are just as (or even more) inclined to share an article to mock or debunk it as they are to commend it – as Express readers note, to their fury, when Remoaners "hijack" their leading-question anti-Europe polls.
Flailing around for an explanation for what it could only view as the election disaster, the Mail has attacked virtually every other source of information, with the "biased" BBC and "uncontrolled" tech giants at the top of the list. But that is to ignore the fact that the Mail is a web giant too – Mail Online is the most widely read news website in the world.
The veteran political journalist Nicholas Jones – who thought the reporting of the election was the vilest of his lifetime - was among several writers who saw the result as proof that the tabloids have become paper tigers whose roar doesn't reach the ears of the young but merely reinforces the views of an ageing readership. In a blog post a few days after the election, he described May as the victim of her tabloid champions, trapped in the echo chamber of a phoney propaganda war.
That's an interesting point. Social media is frequently described as an echo chamber; we see the posts of people whose opinions we share. But so it is, too, with newspapers. The Sun is a Tory paper with many Labour-voting readers, but the Express, Mail and Telegraph are all preaching to the choir. Their readers are always going to vote Conservative. Whose minds are they going to change?
Setting the agenda
In an interview with the New York Times a couple of months ago, Tony Gallagher seemed to backtrack on his crowing Brexit text about the power of the Press. This time, he suggested that its influence was limited, residing in its capacity to set the news agenda - in that broadcasters reviewed the papers and followed up their stories.
But that is hardly limited. It means they are reaching an audience far beyond their readership – and their screaming headlines are seen in shops everywhere by people who will never part with their money to read the full article.
Professor Steven Barnett of Westminster University agrees with Gallagher on newspapers' ability to set the news agenda, but believes their influence goes beyond that, not least because their writers appear as pundits on political and current affairs programmes.
Newspapers can also devote resources to a brand of journalism untrammelled by the rules of impartiality that bind broadcasters, so they can pursue particular stories to suit a political agenda. Finally, Barnett cites research showing that people still pay far more attention to material read in hard copy than on screen.
Denied at the last?
Further evidence that papers still have clout comes from within the Labour party. One activist's experience on the doorsteps, coupled with analysis of post-election polling data, has led him to believe that had it not been for the "incredible assault from the right-wing press" in the last few days of the campaign, Corbyn would now be in No 10.
Pete Heskett, director of the marketing and strategy company Hatch the Plan, noted a huge swing towards Labour through the campaign that suddenly reversed just before the vote. That coincided with a "massive bounce back" for May's rating over Corbyn on who would make the best prime minister. "I can't believe this was the result of any great performance from her. Her reaction to the London Bridge terror attack was to politicise the atrocity, but I don't believe that this would have been enough to overcome her poor performance in the campaign.
"It wasn't that May's rating stabilised, it was that Corbyn's absolutely tanked in the last couple of days, actually hitting a lower level than it had been pre-campaign, when everyone said he was a dead duck. This I believe is the data that highlights that newspapers did, in fact, have a decisive impact on the election."
Would Corbyn have got across the line without that all-out assault? The chances are we'll be back in the autumn for a re-run and maybe we'll find out then. One thing is sure: whoever leads the Tory party will have the unswerving support of the Express (in the space of a week last summer, it lined up behind Gove, Johnson, Leadsom and May), the Mail and, most likely, the Sun.
They've already started as they mean to go on – essentially bash Corbyn and find "Look over there!" stories when the compelling news is inconvenient.
The Queen in mufti for the state opening of parliament is by any standards a shoo-in front-page picture, but was strangely absent from the Three Brexiteers' covers. Could it have been that "EU flag" hat? They must have heaved a sigh of relief at the Express when she popped back to the Palace to change into bright yellow for Ascot.
And that £1bn deal with the DUP to keep May in power?
Who cares when you can write about leftie school heads, lesbian jail romps or a diabetes cure?
Feature National/international news media
IFJ welcomes lowest number of killings of journalists for a decade
SoE welcomes press freedom funding
Newsworks announces 2018 Planning Awards shortlist
Brexit and discrimination in the UK press
Mail on Sunday to launch new lifestyle section
European Digital Media Awards – winners announced
Sun Savers appoints first Savers Editor
Trinity Mirror: Appointment of Non Executive Director & Chairman Designate
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4 Hearst Magazines in Europe unveils new structure 9 Gadgets, Health, Smart Living & Celebs top the Readly 2019 Review
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2 Time Out appoints new CMO 7 Time Out announces new London Editor and Head of Digital Content
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